Hamiltonian systems, symplectic flows, classical integrable systems

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31 views

### Relation between symplectic blow-up of a compact manifold and fibre bundles over same manifold

The symplectic blow-up of a compact symplectic manifold $(X,ω)$ along a compact symplectically embedded submanifold $(M,σ)$ results in another compact manifold $(\tilde{X},\tilde{ω})$ given by
$$\...

**1**

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47 views

### Shape of the bubbling limit of holomorphic discs

I will present my question in the specifics I encountered it, so maybe some of the details are irrelevant for the desired conclusion.
Consider $(S^2\times S^2,\omega_{std})$ the product of two ...

**6**

votes

**2**answers

299 views

### Atiyah-Guillemin-Sternberg convexity theorem

I would like to study the Atiyah-Guillemin-Sternberg convexity theorem: proof and applications. I am already familiarised with hamiltonian actions, moment maps...and with elementary Morse theory.
So ...

**1**

vote

**1**answer

117 views

### Are symplectic realizations of a Poisson manifold unique?

If $(M, P)$ is a (Hausdorff) Poisson manifold, then there exist a surjective Poisson submersion $\pi : (S, \omega) \to (M, P)$ with $(S, \omega)$ a symplectic manifold.
I am in a situation where I ...

**6**

votes

**1**answer

98 views

### Sign problem in a Calogero-Moser system: proof of integrability?

Everyone of us had sometimes this awful feeling that some sign is lost in a calculation and that this sign is perturbing some fundamental understanding of what is going on. I feel the same has ...

**3**

votes

**1**answer

294 views

### Why curvature is equivariant as a moment map?

In Atiyah and Bott's famous paper "The Yang-Mills Equations over Riemann Surfaces", they treated curvature as a moment map of the gauge group acts on the connection space of a principal bundle $P$ ...

**5**

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**1**answer

122 views

### Ascending surfaces in the 4-ball

Let the standard symplectic structure on $B^4$ (viewed in $\mathbb{R}^4$ or $\mathbb{C}^2$) be given by $\omega=(1/2) d \eta$, for
\begin{align*}
\eta &:=
x_1 \, dy_1 - y_1 \, dx_1 + x_2 \, ...

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225 views

### Weil-Petersson metric is quasi isometric with which model?

Let $\mathcal M_g$ be the moduli space of curves of genus $g$. If we take $X^{reg}=X\setminus D$, where D is a divisor with normal crossings. Endow
$X^{reg}$ with a complete Kahler metric which has a ...

**0**

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66 views

### G-invariant functions on manifold for G compact

In a paper I saw the following statement:
Let $M$ be a connected symplectic manifold and $G$ be a compact Liegroup acting symplectically and hamiltonian on $M$. Let $\Phi \colon M \to \mathfrak{g^*}$ ...

**3**

votes

**1**answer

101 views

### Duistermaat-Heckman integral formula on compact manifold with boundary

Let a compact Lie group $G$ acts on a closed symplectic manifold $(M,\omega)$. If the action is Hamiltonian with $\mu$ the moment map, then the integral $$\int_M e^{i\mu (X)+\omega}$$ is equal to the ...

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135 views

### Ricci flat metric on pair (X,D)

Let $(X,\omega)$ be a Calabi-Yau variety and $D$ be a simple normal crossing divisor on $X$ with conic singularities with cone angle $2\pi\theta$, $0<\theta<1$ such that $K_X+D>0$, then is ...

**5**

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74 views

### Uniqueness of Birkhoff Normal Form and KAM theory for Symplectomorphims

I am starting to work with Hamiltonian Dynamics and I have been taking a look at some of the basic stuff in KAM theory. I have posted part of this question at MSE but as I did not get any response I ...

**3**

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**0**answers

58 views

### Delzant polytopes and combinatorial types

At first, let us see the following matheoverflow question,
About a Delzant polytope. (In particular dodecahedron)
The question is whether (combinatorial) regular dodecahedron can be realized as a ...

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votes

**0**answers

176 views

### Integrations of Ricci curvature of the Weil-Petersson metric on the moduli space of varieties of general type is a rational numbers?

It is known that the integrations of Ricci curvature of the Weil-Petersson metric on the moduli space of Calabi-Yau varieties is a rational numbers
My question is about the moduli space of ...

**3**

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**0**answers

92 views

### Fibers of torus equivariant moment maps

Given a closed (possibly singular) projective variety $V$ with a symplectic structure and a torus action, there is a moment map
$\mu: V \rightarrow Lie(T)^*$. Note that the dimension of $T$ could be ...

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88 views

### What is this 2-form on a Lagrangian torus fibration?

Suppose we are given a regular $2n$-dimensional Lagrangian fibration $\pi : (M,\omega) \to B$ with connected, compact fibers. Then it is well-known (Arnold-Liouville) that each fibre is a Lagrangian ...

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**2**answers

686 views

### What is the obstruction to the existence of a global Kahler potential?

It is a well-know fact that if $(X,\omega)$ is Kahler then about every point $x \in X$ there exists a neighbourhood $U$ and a function $K \in C^{\infty}(U,\mathbb{R})$ such that $\omega|_U = i\partial ...

**2**

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**0**answers

400 views

### Weil Petersson metric on moduli space of Calabi Yau manifolds

Let $f:(X,D)\to Y$ be a holomorphic fibre space where $D$ is divisor with conic singularities and let fibres $(X_s,D_s)$ are log Calabi-Yau pair .i.e $K_X+D$ is nummerically trivial, then we have ...

**2**

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**0**answers

107 views

### holomorphic curves in almost toric fibration and their relation to tropical curves

My goal is to get better understanding how the projection of holomorphic curves converge to tropical disks.
We are given an almost toric fibration $X\rightarrow B$ with special Lagrangian fibers with ...

**2**

votes

**2**answers

372 views

### $dd^\mathbb{C}$-lemma on pair $(X,D)$

Let $X$ be a Kähler manifold with a simple normal crossing divisor $D$, i.e., pair $(X,D)$. Let $\omega$ and $\omega'$ be two Kähler forms in the same Kähler class then have we $dd^\mathbb{C}$-lemma ...

**2**

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**0**answers

122 views

### Flow on invariant Lagrangian tori

The most concrete version of the question is :
A (necessarily) invariant Lagrangian torus $L$ on the unit cotangent of a Riemannian metric on the two-torus carries a periodic orbit with period $T$. ...

**3**

votes

**1**answer

133 views

### Condition on moment polytope for a toric manifold to be Fano

Suppose $M$ is a symplectic toric manifold. This means there is a compact torus
$T$ that has a Hamiltonian action on $M$, with moment map $\mu:M \to \mathfrak t^*$, and $\dim(M)=2\dim(T)$. Can one ...

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**0**answers

166 views

### Handle attachment in symplectic category

It is known that for an exact symplectic manifold $(M,\omega_M)$ with a convex boundary $(\partial M,\theta_M)$, where $d\theta_M=\omega_M$ (usually called a Liouville domain), one can attachment to ...

**11**

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**1**answer

277 views

### Pseudo-holomorphic curves in the six-sphere

Equip $S^6$ with the almost complex structure coming from the cross product on $\mathbb R^7$ (i.e. the product on the pure imaginary octonions). What is known about the psudo-holomorphic curves in ...

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44 views

### Is the preimage of coisotropic submanifold coisotropic?

Let $M$ and $N$ be smooth manifolds with Poisson-structures $\{ \cdot , \cdot\}|_M$ and $\{\cdot , \cdot \}|_N$ We call $\phi: M \to N$. a Poisson-map, if the pullback of $\phi$ is compatible with the ...

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**0**answers

122 views

### From symplectic manifold to Hilbert spaces [closed]

What could be a mathematical model of such physical wish? I'm looking for something sending a symplectic manifold $(M,\omega)$ to a Hilbert space $H_{M}$ with the following properties:
1- We should ...

**13**

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**0**answers

332 views

### What is the Hochschild cohomology of the Fukaya-Seidel category?

Let $(Y, \omega)$ be a compact symplectic manifold and let $Fuk(X,\omega)$ be its Fukaya category. The Hochschild cohomology of this category should be given by $HH^\bullet(Fuk(Y,\omega))=H^\bullet(Y, ...

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280 views

### Question about theorem in Arnold's book on action-angles variables

I have a question about the action-angle theorem on p. 283 in Arnold's textbook on classical mechanics.(I added the link to this book in the last part of this question)
If you don't have the book or ...

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108 views

### Is there an algorithm to compute the intersection of tautological classes on the moduli space of genus one curves?

Let $\overline{M}_{1,1}(\mathbb{P}^2, d) $ be the moduli space of degree
$d$ genus one curves on $\mathbb{P}^2$ with one marked point. Let
$L\longrightarrow \overline{M}_{1,1}(\mathbb{P}^2, d) $ ...

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votes

**1**answer

103 views

### Unique Equivariant Symplectic Structure for the Full Flag Manifold of $SU(3)$?

I was looking at the following interesting question about the number of equivariant almost complex structures on the full flag manifold of $SU(3)$, and I began to wonder how many equivariant ...

**3**

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**1**answer

158 views

### Local symplectomorphisms become global ones?

It is widely known that a local diffeomorphism is not necessarily a global diffeomosphism and so on.
Now, I stumbled over the question whether in some particular cases, as I will describe below, ...

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**2**answers

207 views

### Blowup and Delzant Polytope

Can someone please explain why blow up in a Symplectic toric manifold corresponds to chopping off a corner in the Delzant polytope?

**5**

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**1**answer

192 views

### Does pseudo-holomorphic *submanifolds* satisfy unique continuation?

Let $f,g:(D^2,j_\mathrm{std})\to(B^{2n}(1),J)$ be two pseudo-holomorphic maps. The following unique continuation result is well-known (it may be proved using either Aronszajn's Lemma or the Carleman ...

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**0**answers

188 views

### Is tightness decidable?

Given a contact structure on a three-manifold, is there an algorithm to decide whether or not it tight?
For concreteness' sake, let's agree to represent the given contact three-manifold via an open ...

**2**

votes

**1**answer

177 views

### How can we define constant scalar curvature Kahler or cscK on pair $(X,D)$

A Kahler metric $\omega$
with cone singularities along divisor
$D$
with cone angle $2\pi\beta$
is said to be
of
constant scalar curvature Kahler
or
cscK
if its scalar curvature $S(\omega)$, which is ...

**2**

votes

**0**answers

134 views

### Poisson Manifold Structures on Even Dimensional Spheres

The $2n$-sphere, for $n=1,2,3$, possess a (non-trivial) Poisson manifold structure. Is this still true for $n > 3$? Describing the spheres as homogeneous spaces $SO(n)/S(n-1)$, are there Poisson ...

**0**

votes

**1**answer

260 views

### Steps in paper on sympl. geometry unclear

I am currently reading a paper on symplectic geometry: Periodic orbits for Hamiltonian systems in cotangent bundles by Christopher Golé.
It deals with the question how the stability properties of a ...

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**0**answers

80 views

### Poisson algebra automorphisms of a symplectic manifold

Let $(M,\omega)$ be a symplectic manifold. Let $V=\mathcal{C}^\infty(M)$ be the Lie algebra of smooth real valued functions. Suppose $f:\rightarrow V$ be an Lie-algebra isomorphism (an algebra ...

**3**

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**1**answer

139 views

### Symplectic geometry and stability of orbits

I am looking for a theorem (read it once, but forgot about it and would now like to find a reference with proof):
In symplectic geometry (at least for some particular subset of $\mathbb{R}^2$), there ...

**3**

votes

**1**answer

133 views

### Action generated by geodesic flow is hamiltonian

I'm trying to understand why a certain action of a Lie Group is hamiltonian.
Let $(M,g)$ be a geodesically complete Riemannian manifold.
Then there exists a canonical one-form on the cotangentbundle $...

**4**

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**1**answer

189 views

### Check symplectomorphism property on infinitesimal generators

I stumbled over the following question:
First, let me give the basic definition of a symplectic group action:
Let $(M, \omega)$ be a symplectic manifold and $G$ a Lie group. A smooth action $\Phi:G \...

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**0**answers

152 views

### Is the complex structure on a del-Pezzo surface a regular complex structure?

Let $(X, \omega, J)$ be a compact symplectic manifold with an almost complex structure. Fix some homology class $\beta \in H_2(X, \mathbb{Z})$. An almost
complex structure $J$ is said to be $\textit{...

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votes

**2**answers

256 views

### Legendrian knot in 3-sphere

We are given a Legendrian knot, fixed up to Legendrian isotopy, in $(S^3,\xi)$ ($\xi$ is the standard contact structure). Does it necessarily bound a symplectic surface in $(B^4,\omega)$ (again $\...

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**1**answer

160 views

### Hamiltonian flow local diffeomorphism?

I am currently reading Arnold's proof of the Darboux theorem in his book on classical mechanics and fail to understand some point.
The background
So he wants to show that any symplectic form is ...

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148 views

### Symplectic spectrum

I have a question about a step in the proof of the following theorem from symplectic geometry.
The theorem is: Given any ellipsoid $E:=\{ w \in \mathbb{R}^{2n}; \sum_{i,j =1}^{2n} a_{ij}w_iw_j \le 1\}...

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73 views

### Limiting behaviour of the symplectic form

It is well known that the coadjoint orbits of the Heisenberg group (with a suitable choice of coordinate system) are the planes $z=c\ne 0$ parallel to the $xy$-plane, and the points in the $xy$-plane. ...

**5**

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**1**answer

99 views

### multiplicity free chain of Lie subgroups gives a completely integrable system on symplectic manifold

Assume we have a symplectic manifold $N$ and a Hamiltonian action of a Lie group $G$, such that the algebra of all $G$-invariant functions on $N$ is commutative under Poisson bracket.
Furthermore, ...

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87 views

### An algebraics Hamiltonian vector field with a finite number of periodic orbits(2)

Is there a polynomial Hamiltonian $H:\mathbb{R}^{4}\to \mathbb{R}$ such that the number of nontrivial periodic orbits of the corresponding Hamiltonian vector field $X_{H}$ is finite but different ...

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293 views

### Proof of Arnold-Liouville theorem in classical mechanics [closed]

I am currently reading Arnold's book "Mathematical Methods of classical mechanics" on page 278 and I don't see through his arguments there at a point.
Especially, I am talking about the part that ...

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**1**answer

167 views

### Non Hamiltonian vector field

Let $\Phi: G \times M \rightarrow M$ be a group action on a symplectic manifold $M$ and $G$ be a Lie group.
Furthermore, $x$ is a solution of the Hamilton equation $\dot{x}(t) = X_H(x(t))$ and for a ...